Homework 2

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.

Answer the following:

  1. You tried converting $\sqrt[5]{x^3}$ into $x^{\frac{5}{3}}$. Is this wrong?
  2. Simplify the following:
    1. $16^{\frac{1}{4}}$
    2. $-8^{\frac{1}{3}}$
    3. $23^{\frac{2}{3}}\cdot 23^{\frac{1}{3}}$
    4. $\left(\dfrac{1}{32}\right)^{-\frac{2}{5}}$
  3. Rewrite each root as an exponent and fully simplify.
    1. $\dfrac{\sqrt[3]{x^2}}{x^{2/3}}$
    2. $\dfrac{(-3)^4\sqrt{x}}{3^2\sqrt[3]{x}}$
    3. $\sqrt[4]{16x^8}$
    4. $\sqrt[4]{x^3}\cdot \sqrt{x}$
    5. $\dfrac{\sqrt[3]{(2x+1)^2}(x-2)^{2/5}}{\sqrt[4]{(2x+1)^5}}$
  4. Your friend tries to simplify an expression. Circle the error and explain to your friend why it is an error. \[\left(\dfrac{x}{2(x-1)}\right)^4 = \dfrac{x^4}{2^4(x-1)^4} = \dfrac{x^4}{16(x^4 - 1^4)} = \dfrac{x^4}{16(x^4 - 1)}\]
  5. State whether each pair of expressions are like terms or not.
    1. $3x^2$ and $4y$
    2. $3(x-1)x$, $4x(x-1)$ and $-x(x-1)$
    3. $-222x^3y$ and $4x^3y$
    4. $5(x+1)(x+2)$ and $-(x+1)(x+2)$
    5. $-100(3x-2)(4x^2+3)^2$ and $4(4x^2+3)^2(3x-2)$
  6. When asked to "expand" a problem, what you are converting from and to?
  7. When subtracting $2$ or more terms, what do you need to not forget around the terms?
  8. When multiplying $2$ or more terms, what do you need to not forget around the terms?
  9. Expand and simplify each expression.
    1. $(2x^2 + 3x) + (3x^3 + 2x)$
    2. $(2x+1)(3x-2)$
    3. $(x^2 + 2x + 1)(x-2)$
    4. $(4x - 3y)(4x + 3y)$
    5. $2(x + h)^2 - 2x^2$
      Hint $(x+h)^2$ is not equal to $x^2 + h^2$. Expand it properly.
    6. $(x + h) - (x + h)^2$
      Hint Remember that the $- (x+h)^2$ is subtracting $\geq 2$ terms. Expand it properly.
    7. $2x(x-3) - (2x - 1)(x + 2)$
    8. $2(x + 1)(x - 2) - 3(x + 1)$
    9. $2(2x-1)^2 - (2x-1)2x$
    10. $2x(x^2 - 1) - 2x(x^2 + 1)$
  10. If \[f = x^2 - x \qquad g = 2x^3 - x + 1\] Expand the following:
    1. $f - g$
    2. $g - f$
    3. $f \cdot g$
  11. If \[a = x - 1 \qquad b = x + 1 \qquad c = x^2 - 1\] Expand the following:
    1. $-(a - b)$
    2. $a\cdot b \cdot c$
    3. $a^2 - b\cdot c$
      Hint $a^2$ means all of $a$ is squared. So $a^2 = (x-1)^2$.
  12. When asked to "factor" a problem, what you are converting from and to?
  13. Fill in this table:
    Number of Terms Factoring Methods to Try (in order)
    2 terms
    3 terms
    4 terms
    $\geq$ 5 terms
  14. Factor the following expressions.
    1. $-2x^3 - x^2$
    2. $4x^2y^3 - 16x^6y^4 + 6xy$
    3. $2(2x-1)^2 - (2x-1)2x$
    4. $2(x + 1)(x - 2) - 3(x+2)(x + 1)$
    5. $(x+3)^2(x-2) - (x+3)(x-2)^2$
    6. $x^2 + 5x + 6$
    7. $x^2 + 13x + 12$
    8. $2x^2 + 7x + 3$
    9. $(x^2 + 1)^2 - 7(x^2 + 1) + 10$
    10. $x^2 + 4x + 4$
    11. $x^2 + 6x + 9$
    12. $4a^2 - 9b^2$
    13. $2x^3y + 8x^2y - 2xy - 8y$
    14. $x^4 - 1$
      Hint Let $A = x^2, B = 1$.
    15. $2x^2yz + 7xyz + 3yz$
  15. When dealing with a fractional expression, what does the word "simplify" mean?
  16. Completely simplify.
    1. $\dfrac{4x^2 + 8x - 12}{2x - 2}$
    2. $\dfrac{x^2 + 2x + 1}{x^2 - 1}$
    3. $\dfrac{x^2h + 2xh + h}{h}$

      4. The remaining problems will be on next week's homework. Skip for now.

    4. $\dfrac{(x+h)^2 - x^2}{h}$
    5. $\dfrac{(x+h)^2 - (x + h) - (x^2 - x)}{h}$
    6. $\dfrac{x^2 + 7x + 12}{x^2 + 3x + 2} \cdot \dfrac{x^2 + 5x + 6}{x^2 + 6x + 9}$
    7. $\dfrac{2x^2 - x - 6}{4x^2 - 7x + 3} \div \dfrac{2x^2 + 5x + 3}{x^2 - 4x + 3}$
    8. $x^{-1} + \dfrac{1}{x - 1}$
    9. $\dfrac{x}{x^2 + 2x + 1} - \dfrac{1}{x + 1}$
    10. $\dfrac{x - 2}{x^2 + 5x + 6} - \dfrac{x + 2}{x^2 + 4x + 3}$